I think the synagogue of Satan actually refers to false Christians and not to false Jews. The synagogue of Satan is mentioned two times in the Book of Revelation, in short letters that John addressed to the churches of Smyrna and Philadelphia. In a third letter addressed to the church of Ephesus, there is a parallel passage where the phrase "those who claim to be apostles but are not" is used in place of the phrase "those who say they are Jews and are not":
- Revelation 2:1-2: "To the angel of the church in Ephesus write: These are the words of him who holds the seven stars in his right hand and walks among the seven golden lampstands: I know your deeds, your hard work and your perseverance. I know that you cannot tolerate wicked men, that you have tested those who claim to be apostles but are not, and have found them false."
- Revelation 2:8-9: "To the angel of the church in Smyrna write: These are the words of him who is the First and the Last, who died and came to life again. I know your afflictions and your poverty - yet you are rich! I know the slander of those who say they are Jews and are not, but are a synagogue of Satan."
- Revelation 3:7-9: "To the angel of the church in Philadelphia write: These are the words of him who is holy and true, who holds the key of David. What he opens no one can shut, and what he shuts no one can open. I know your deeds. See, I have placed before you an open door that no one can shut. I know that you have little strength, yet you have kept my word and have not denied my name. I will make those who are of the synagogue of Satan, who claim to be Jews though they are not, but are liars - I will make them come and fall down at your feet and acknowledge that I have loved you."
Therefore the word Jew might have a similar meaning as the word apostle, so that it refers to a spiritual Jew or a fellow believer (
https://cgg.org/index.cfm/library/bqa/id/114/what-is-synagogue-satan-revelation-29-39.htm):
> The word _synagogue_ comes from a Greek word meaning "assembly of men" or "congregation," and it was used much like the English word "church." The "synagogue of Satan," then, is an assembly or congregation - a church - made up of the individuals who "say they are Jews, and are not." The term "Jew" is used here in a spiritual sense.
> Notice the apostle Paul's definition of a spiritual Jew in Romans 2:29: "He is a Jew who is one inwardly, and circumcision is that of the heart, in the Spirit, and not in the letter; whose praise is not from men but from God." In this case, it is not one's physical race that counts, but his spiritual condition (Galatians 3:27-29; Romans 4:16). True Christians are spiritual Jews because Jesus says in John 4:22, "Salvation is of the Jews."
> Those in the "synagogue of Satan" say they are spiritual Jews - pretend to be real Christians - but are not. This false church was already developing in the days of the apostle John, masquerading as God's true church. It had congregations in the cities of Smyrna and Philadelphia in Asia Minor even at that early time. It and its daughter churches are further described in Revelation 17.
The concept of a false apostle is also mentioned in 2 Corinthians 11, where Paul says that the false apostles are servants of Satan, where he says that the false apostles are Christians who preach a different version of the gospel than himself, and where he says that like himself, the false apostles are Hebrews and they are Israelites: "For if someone comes to you and preaches a Jesus other than the Jesus we preached, or if you receive a different spirit from the one you received, or a different gospel from the one you accepted, you put up with it easily enough. But I do not think I am in the least inferior to those 'super-apostles.' [...] For such men are false apostles, deceitful workmen, masquerading as apostles of Christ. And no wonder, for Satan himself masquerades as an angel of light. It is not surprising, then, if his servants masquerade as servants of righteousness. [...] Are they Hebrews? So am I. Are they Israelites? So am I. Are they Abraham's descendants? So am I."